Theoretical study of multiple equilibria in simple axisymmetric tropical circulations

Abstract

Possible existence of multiple equilibria in axisymmetric circulation is examined using simple models. A simple two-layer axisymmetric model is constructed using balance equations on an equatorial beta-plane. The model has no mountains and the thermal forcing is represented by a Newtonian cooling law. A radiative equilibrium temperature distribution is prescribed that has a maximum around 25° latitude, intended to simulate the summer conditions in the northern hemisphere. The effect of heating due to condensation is included only through the change in the static stability parameter. A host of steady state solutions are obtained analytically from a severely truncated version of this model. An analysis of stability of these steady states with respect to first and second y-mode perturbations reveal that only one steady state is stable against both types of perturbation. Two other steady states are found to be quasi-stable (e-folding time greater than three weeks) with respect to these perturbations. We then examined two different simple numerical models for possible existence of multiple equilibria in symmetric circulations. First, the time dependent spectral equations for the above two-layer model, with many more degrees of freedom, were numerically integrated with various initial conditions given by the steady state solutions for the severely truncated model. These integrations show that the model gives only one steady state corresponding to the stable state given by the truncated model. Secondly, a multi-level grid point model developed by Kalnay-Rivas (1973) was examined. This model uses primitive equations under Boussinesq approximations. Parameterizations for thermal forcing and dissipation used in this model were similar to those used in the spectral model. It is found by integrating this model for various initial conditions that this model also gives only one steady state similar to the steady state obtained by the spectral model. The inability of these simple models to exhibit multiple equilibria in axisymmetric circulations is discussed.